The Polya–Schur characterization for 0-sum hyperbolicity preservers

Let $T:\mathbb{R}[t]_{n,0}\to\mathbb{R}[t]_{d,0}$ be a diagonal linear map. A 0-sum hyperbolicity preserver is a map preserving the relevant 0-sum hyperbolicity property. Polya–Schur conjecture. $T$ is a 0-sum hyperbolicity preserver if and only if

$$T\bigl((x-1)^{n-1}(x+n-1)\bigr)$$

has real roots, with $d-1$ of those roots having the same sign. If true, this would characterize all hook-shaped symmetric hyperbolic polynomials. The conjecture is proved for $d\leq 4$, its sign condition is necessary for all $d$, and computational evidence supports it for $d\leq 6$; the general case remains open.